gives the Kelvin function TemplateBox[{z}, KelvinBer].


gives the Kelvin function TemplateBox[{n, z}, KelvinBer2].


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For positive real values of parameters, TemplateBox[{n, z}, KelvinBer2]= Re(e^(npii)TemplateBox[{n, {z, , {e, ^, {(, {{-, pi}, , {i, /, 4}}, )}}}}, BesselJ]). For other values, is defined by analytic continuation.
  • KelvinBer[n,z] has a branch cut discontinuity in the complex z plane running from to .
  • KelvinBer[z] is equivalent to KelvinBer[0,z].
  • For certain special arguments, KelvinBer automatically evaluates to exact values.
  • KelvinBer can be evaluated to arbitrary numerical precision.
  • KelvinBer automatically threads over lists.


open allclose all

Basic Examples  (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope  (36)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (3)

Values at zero:

Find the first positive minimum of KelvinBer[0,x]:

For half-integer orders, KelvinBer evaluates to elementary functions:

Visualization  (3)

Plot the KelvinBer function for integer () and half-integer () orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

Function Properties  (13)

The real domain of TemplateBox[{0, x}, KelvinBer2]:

The complex domain of TemplateBox[{0, x}, KelvinBer2]:

TemplateBox[{{-, {1, /, 2}}, x}, KelvinBer2] is defined for all real values greater than 0:

The complex domain is the whole plane except :

Approximate function range of TemplateBox[{0, x}, KelvinBer2]:

Approximate function range of TemplateBox[{1, x}, KelvinBer2]:

TemplateBox[{0, x}, KelvinBer2] is an even function:

TemplateBox[{1, x}, KelvinBer2] is an odd function:

KelvinBer threads elementwise over lists:

TemplateBox[{0, z}, KelvinBer2] is an analytic function of :

KelvinBer is neither non-decreasing nor non-increasing:

KelvinBer is not injective:

KelvinBer is neither non-negative nor non-positive:

TemplateBox[{n, z}, KelvinBer2] has singularities or discontinuities in the non-positive reals when is not an integer:

KelvinBer is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

The first derivative with respect to z:

The first derivative with respect to z when n=1:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the derivative with respect to z:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

The definite integral:

More integrals:

Series Expansions  (5)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The general term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

The Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

Functional identity:

Recurrence relations:

Generalizations & Extensions  (1)

KelvinBer can be applied to a power series:

Applications  (3)

Solve the Kelvin differential equation:

Plot the resistance of a wire with circular cross section versus AC frequency (skin effect):

This MeijerG is simplified to KelvinBei and KelvinBer functions:

Properties & Relations  (4)

Use FullSimplify to simplify expressions involving Kelvin functions:

Use FunctionExpand to expand Kelvin functions of half-integer orders:

Integrate expressions involving Kelvin functions:

KelvinBer can be represented in terms of MeijerG:

Possible Issues  (1)

The oneargument form evaluates to the two-argument form:

Wolfram Research (2007), KelvinBer, Wolfram Language function,


Wolfram Research (2007), KelvinBer, Wolfram Language function,


Wolfram Language. 2007. "KelvinBer." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). KelvinBer. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_kelvinber, author="Wolfram Research", title="{KelvinBer}", year="2007", howpublished="\url{}", note=[Accessed: 15-April-2024 ]}


@online{reference.wolfram_2023_kelvinber, organization={Wolfram Research}, title={KelvinBer}, year={2007}, url={}, note=[Accessed: 15-April-2024 ]}